Input-Power-to-State Stability of Time-Varying Systems

TL;DR

This paper introduces input-power-to-state stability (IPSS), a new stability concept for time-varying systems where traditional ISS and iISS are insufficient, providing necessary and sufficient conditions under mild assumptions.

Contribution

It develops the theoretical foundation for IPSS, linking it to ISS-Lyapunov functions and iISS, and offers a converse Lyapunov theorem for systems with possibly discontinuous dynamics.

Findings

Dissipation-form ISS-Lyapunov functions imply IPSS.

iISS with exponential class-\KL{} functions implies IPSS.

ISS with stronger assumptions guarantees IPSS.

Abstract

When the state of a system may remain bounded even if both the input amplitude and energy are unbounded, then the state bounds given by the standard input-to-state stability (ISS) and integral-ISS (iISS) properties may provide no useful information. This paper considers an ISS-related concept suitable in such a case: input-power-to-state stability (IPSS). Necessary and sufficient conditions for IPSS are developed for time-varying systems under very mild assumptions on the dynamics. More precisely, it is shown that (a) the existence of a dissipation-form ISS-Lyapunov function implies IPSS, but not necessarily that of an implication-form one, (b) iISS with exponential class-$\KL$ function implies IPSS, and (c) ISS and stronger assumptions on the dynamics imply the existence of a dissipation-form ISS-Lyapunov function and hence IPSS. The latter result is based on a converse Lyapunov theorem for time-varying systems whose dynamics (i.e. state derivative) is not necessarily continuous with respect to time.